On the Strong Metric Dimension of Annihilator Graphs of Commutative Rings

For a connected graph G ( V ,  E ), a vertex $$w\in V(G)$$ w ∈ V ( G ) strongly resolves two vertices $$u, v \in V(G)$$ u , v ∈ V ( G ) if there exists a shortest $$u-w$$ u - w path containing v or a shortest $$v-w$$ v - w path containing u . A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S . The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G . Let R be a commutative ring with identity, and let Z ( R ) be the set of zero-divisors of R . The annihilator graph of R is a simple graph with the vertex set $$Z(R)^*=Z(R){\setminus }\{0\}$$ Z ( R ) ∗ = Z ( R ) \ { 0 } , and two distinct vertices x and y are adjacent if and only if $$ann_R(xy)\ne ann_R(x)\cup ann_R(y)$$ a n n R ( x y ) ≠ a n n R ( x ) ∪ a n n R ( y ) . In this paper, we study the strong metric dimension of annihilator graphs associated with commutative rings and some strong metric dimension formulae for annihilator graphs are given.